Solve for $x$ : $ 3|x + 10| + 7 = -5|x + 10| + 10 $
Solution: Add $ {5|x + 10|} $ to both sides: $ \begin{eqnarray} 3|x + 10| + 7 &=& -5|x + 10| + 10 \\ \\ { + 5|x + 10|} && { + 5|x + 10|} \\ \\ 8|x + 10| + 7 &=& 10 \end{eqnarray} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} 8|x + 10| + 7 &=& 10 \\ \\ { - 7} &=& { - 7} \\ \\ 8|x + 10| &=& 3 \end{eqnarray} $ Divide both sides by ${8}$ $ \dfrac{8|x + 10|} {{8}} = \dfrac{3} {{8}} $ Simplify: $ |x + 10| = \dfrac{3}{8}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 10 = -\dfrac{3}{8} $ or $ x + 10 = \dfrac{3}{8} $ Solve for the solution where $x + 10$ is negative: $ x + 10 = -\dfrac{3}{8} $ Subtract ${10}$ from both sides: $ \begin{eqnarray} x + 10 &=& -\dfrac{3}{8} \\ \\ {- 10} && {- 10} \\ \\ x &=& -\dfrac{3}{8} - 10 \end{eqnarray} $ Change the ${ - 10}$ to an equivalent fraction with a denominator of $8$ $ x = - \dfrac{3}{8} {- \dfrac{80}{8}} $ $ x = -\dfrac{83}{8} $ Then calculate the solution where $x + 10$ is positive: $ x + 10 = \dfrac{3}{8} $ Subtract ${10}$ from both sides: $ \begin{eqnarray} x + 10 &=& \dfrac{3}{8} \\ \\ {- 10} && {- 10} \\ \\ x &=& \dfrac{3}{8} - 10 \end{eqnarray} $ Change the ${ - 10}$ to an equivalent fraction with a denominator of $8$ $ x = \dfrac{3}{8} {- \dfrac{80}{8}} $ $ x = -\dfrac{77}{8} $ Thus, the correct answer is $x = -\dfrac{83}{8} $ or $x = -\dfrac{77}{8} $.